---
title: "Cylindrical shell: stresses, max shear and volume change"
subject: "Strength of Materials"
chapter: 5
key: "semester_3/strength_of_materials:cyl-shell-4m-1m-2nmm2-8mm"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/cyl-shell-4m-1m-2nmm2-8mm
site: BCT Engineering
---

# Cylindrical shell: stresses, max shear and volume change

Strength of Materials · Chapter 5: Thin Walled Vessels

- Asked in: 2066 Bhadra (old course), 8 marks
- Video: [Thin-Walled PRESSURE VESSELS in 8 MINUTES - Mechanics of Materials](https://www.youtube.com/watch?v=ic8LmMUr8js) — Less Boring Lectures
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/cyl-shell-4m-1m-2nmm2-8mm
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-5

## Question

A cylindrical shell 4m long and 1m diameter is subjected to an internal pressure of $2\ \text{N/mm}^2$. If the thickness of the shell is 8mm, find the circumferential stress and longitudinal stress. Find also the maximum shear stress and change in volume.

## Short answer

Circumferential stress σ_c = 125.0 N/mm², longitudinal σ_l = 62.5 N/mm². Max shear stress = 31.25 N/mm². Volumetric strain ε_v = 1.1875×10⁻³ gives δV = 3731 cm³ (increase).

## Answer

**Data:** $L = 4000$ mm, $d = 1000$ mm, $p = 2\ \text{N/mm}^2$, $t = 8$ mm. The change in volume needs $E$ and $\nu$; they are not given, so the usual values for steel are taken: $E = 2\times10^5\ \text{N/mm}^2$ and $\nu = 0.3$.

#### Stresses
$$
\sigma_c = \frac{pd}{2t} = \frac{2(1000)}{2(8)} = 125.0\ \text{N/mm}^2, \qquad \sigma_l = \frac{pd}{4t} = 62.5\ \text{N/mm}^2
$$

#### Maximum shear stress
$$
\tau_{max} = \frac{\sigma_c-\sigma_l}{2} = \frac{125.0-62.5}{2} = 31.25\ \text{N/mm}^2
$$

#### Change in volume
$$
\varepsilon_v = \frac{pd}{4tE}(5-4\nu) = \frac{2(1000)}{4(8)(2\times10^5)}(5-1.2) = 1.1875\times10^{-3}
$$

$$
V = \frac{\pi}{4}(1000)^2(4000) = 3.1416\times10^{9}\ \text{mm}^3, \qquad \delta V = \varepsilon_v V = 3.7306\times10^{6}\ \text{mm}^3 = 3731\ \text{cm}^3
$$

**Answer:** $\sigma_c = 125.0$ N/mm², $\sigma_l = 62.5$ N/mm², $\tau_{max} = 31.25$ N/mm², $\delta V \approx 3731$ cm³ (increase).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Hoop stress for d = 1000 mm, t = 8 mm, p = 2 N/mm^2?
   A. 250 N/mm^2
   B. 125 N/mm^2
   C. 62.5 N/mm^2
   D. 31.25 N/mm^2
   Answer: B. pd/2t = 2 x 1000/16 = 125. (Key point: Hoop stress)

2. Maximum shear stress?
   A. 62.5 N/mm^2
   B. 93.75 N/mm^2
   C. 31.25 N/mm^2
   D. 187.5 N/mm^2
   Answer: C. (125 - 62.5)/2 = 31.25 N/mm^2. (Key point: Shear stress)

3. E and nu are not given for the volume change; what is done?
   A. Volume change cannot be found
   B. Assume nu = 0.5
   C. Ignore E and use p only
   D. Assume steel values E = 2e5 N/mm^2, nu = 0.3
   Answer: D. Volumetric strain needs E and nu, so usual steel values are assumed. (Key point: Missing data)

4. Which formula gives the volume change?
   A. delta V = [pd(5 - 4 nu)/(4tE)] x V
   B. delta V = pd/(2tE) x V
   C. delta V = sigma_c/E x V
   D. delta V = 3 sigma_l/E x V
   Answer: A. Volumetric strain times the original volume, giving about 3731 cm^3. (Key point: Volume change)

## Related in Strength of Materials

- [Thin cylindrical shell: max shear, dimension change](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Athin-shell-shear-dimension-change) — Strength of Materials, chapter 5, asked 2×
- [Thin cylindrical shell: change in length and diameter](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Athin-cyl-shell-4m-25nmm2) — Strength of Materials, chapter 5
- [Thin shell: stresses and dimension changes](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Athin-shell-5m-1m-20mm-5nmm2) — Strength of Materials, chapter 5
- [Cylindrical shell with spherical ends: volume change](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Acyl-shell-spherical-ends-volume-change) — Strength of Materials, chapter 5
- [Cylindrical shell with joint efficiencies: change in dimensions](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Acyl-shell-joint-efficiency) — Strength of Materials, chapter 5
- [Steel spherical vessel: circumferential stress and diameter change](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Asteel-spherical-vessel-1000mm-1mpa) — Strength of Materials, chapter 5

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